where \(\sigma _k[B]\)  is the \(k\) -th singular value of a matrix \(B\) , with the convention that \(\sigma _k[B] \ge \sigma _{k+1}[B].\) 
We also define \(\gamma ^k(1/L,A^{E,\theta ,w})= \sum _{j=1}^k \gamma _j(1/L,A^{E,\theta ,w})\) .
The LEs \(\mathbb {R} \ni t \mapsto \gamma ^k(1/L,A^{E,\theta ,w}(\bullet +it))\)  are then convex and piecewise affine functions [1]}. We emphasize that this property may however not be true for Lyapunov exponents \(\gamma _k(1/L,A^{E,\theta ,w}).\) 
In particular, one has Thouless' formula [2]}
\(\begin{split}\gamma ^4(1/L,A^{E,\theta ,w}) &= \int _{{\mathbb {R}}} \log \vert E-E^{\prime } \vert \, dn_{H_w(\theta ,\phi )}(E^{\prime })\\&= \lim _{N \rightarrow \infty } \frac{\log \vert \operatorname{det}(H_{w,{[0,N]}}(\theta ,\phi ,\vartheta )-E)\vert }{N},\end{split}\) 
